Starting from \((w,v) = (1/4,2)\) , iterates approach \(S_{-,\varepsilon }^a\)  and subsequently follow iterates of the map (REF ) up to a neighbourhood of \(F_-\) , similarly to cases I and II. After leaving a neighbourhood of \(F_-\) , iterates are exponentially attracted to an attracting slow manifold \(S_{+,\varepsilon }^{a,2}\)  which perturbs
from (a suitable submanifold of) the normally hyperbolic and attracting manifold of period-2 points \(S^2\)  in Figure REF ; the existence and properties of the slow manifold \(S_{+,\varepsilon }^{a,2}\)  and its stable manifold follow after applying the results in Section  to the second iterate map \((w,v)^\textnormal {T} \mapsto (\bar{w}^2, \bar{v}^2)^\textnormal {T}\) . Iterates move leftward along \(S_{+,\varepsilon }^{a,2}\)  in a `flip-like' orientation-reversing manner about the repelling slow manifold \(S_{+,\varepsilon }^r\) . They subsequently traverse a neighbourhood of the flip point \(Q\) , before being attracted to \(S_{+,\varepsilon }^a\)  after a delay effect which is described in detail in [1]}. Finally, iterates track iterates of the map (REF ) along \(S_{+,\varepsilon }^a\)  towards to fold \(F_+\) , and return to a neighbourhood of \(S_{-,\varepsilon }^a\)  after leaving a neighbourhood of \(F_+\) . Assuming a simple jump-type dynamics at \(F_\pm \)  (as is indeed observed numerically), the geometric sequence repeats, providing a mechanism for non-chaotic bursting.
